## Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

We give a 2-approximation algorithm for the non-uniform Sparsest Cut problem that runs in time n^{O(k)}, where k is the treewidth of the graph. This improves on the previous 2^{2k}-approximation in time poly(n) 2^{O(k)} due to Chlamtac et al. [18].

To complement this algorithm, we show the following hardness results: If the non-uniform Sparsest Cut has a ρ-approximation for series-parallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than 17/16 - ε for ε > 0; assuming the Unique Games Conjecture the hardness becomes 1/α_{GW} - ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 - ε assuming the Unique Games Conjecture.

Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation.